The invention relates to quantum computing and, more specifically, is a novel optical method for constructing a quantum computer.
Many kinds of numerical problems cannot be solved using conventional computers because of the time required to complete the computation. For example, the computer time required to factor an integer containing N digits is believed to increase exponentially with N. It has been estimated that the time required to factor a 150-digit number using the fastest supercomputers currently available would be longer than the age of the universe. Future increases in the speed of conventional computers will clearly be inadequate for problems of that kind, which are often of considerable practical importance. For example, the difficulty in factoring large numbers forms the basis for the most commonly used methods of cryptography.
It has been shown that quantum-mechanical computers could use nonclassical logic operations to provide efficient solutions to certain problems of that kind, including the factoring of large numbers. As an example of a nonclassical logic function, consider the conventional NOT operation, which simply flips a single bit from 0 to 1 or from 1 to 0. In addition to the usual NOT, a quantum computer could also implement a new type of logic operation known as the square root of NOT. When this operation is applied twice (squared), it produces the usual NOT, but if it is applied only once, it gives a logic operation with no classical interpretation.
In addition to performing nonclassical logic operations, quantum computers will be able to perform a large number of different calculations simultaneously on a single processor, which is clearly not possible for a conventional computer. This quantum parallelism is responsible for much of the increased performance of a quantum computer.
The operation of individual quantum logic gates has been demonstrated, but no operational quantum computer has been constructed. The eventual goal is to produce large numbers of quantum logic gates on a single substrate, in analogy with current semiconductor technology, which would allow the development of quantum computers for practical applications.
Quantum computers will use a binary representation of numbers, just as conventional computers do. An individual quantum bit, often called a qubit, will be physically represented by the state of a quantum system. For example, the ground state of an atom could be taken to represent the value 0, while an excited state of the same atom could represent the value 1. In the optical approach of the invention to quantum computing, a 0 is represented by a single photon in a given path. The same photon in a different path represents a 1.
Although classical bits always have a well-defined value, qubits often have some probability of being in either of the two states representing 0 and 1. It is customary to represent the general state of a quantum system by |"psgr" greater than , and we will let |0 greater than  and |1 greater than  represent the states corresponding to the values 0 and 1, respectively. Quantum mechanics allows superpositions of these two states, given by
|"psgr" greater than =xcex1|0 greater than +xcex2|1 greater than 
where xcex1 and xcex2 are complex numbers. The probability of finding the system in the state |0 greater than  is equal to xcex12 the probability of the state |1 greater than  is xcex22.
Quantum-mechanical superpositions of this kind are fundamentally different from classical probabilities in that the system cannot be considered to be in only one of the states at any given time. For example, consider a single photon passing through an interferometer, as illustrated in FIG. 1, with phase shifts xcfx861 and xcfx862 inserted in the two paths. A beam splitter gives a 50% probability that the photon will travel in the upper or the lower path. If a measurement is made to determine where the photon is located, it will be found in only one of the two paths. But if no such measurement is made, a single photon can somehow measure both phase shifts xcfx861 and xcfx862 simultaneously, since the observed interference pattern depends on the difference of the two phases. This suggests that in some sense a photon must be located in both paths simultaneously if no measurement is made to determine its position. In a more complicated interferometer with many paths, a single photon can simultaneously measure a linear combination of the phase shifts in all of the paths even though it can be detected in only one of the paths.
The ability of a quantum computer to perform more than one calculation at the same time is analogous to the properties of the single-photon interferometer just described. A quantum computer,can provide results that depend on having performed a large number of calculations, even though a measurement to determine exactly what the computer was doing would show that it was programmed to perform only one specific calculation. To illustrate this, consider a computer programmed to perform a specific calculation based on the value of N input bits, and assume that the result can be described by N output bits, as illustrated in FIG. 2. There are 2N different combinations of input bits, each of which corresponds to a specific input state denoted by |inputj greater than , where j takes on all the values from 1 to 2N. The equal number of specific combinations of output bits is denoted by |outputk greater than . Each input state can produce a superposition of possible output states,       "LeftBracketingBar"          input      j        ⟩    →            ∑              k        =        1                    2        N              ⁢          xe2x80x83        ⁢                  β        jk            ⁢              "LeftBracketingBar"                  output          k                ⟩            
where the complex coefficients xcex2jk describe the calculation performed. In addition, the input state can be a superposition of all of the possible inputs to the computer:       "LeftBracketingBar"    input    ⟩    =                    ∑                  j          =          1                          2          N                    ⁢              xe2x80x83            ⁢                        a          j                ⁢                              "LeftBracketingBar"                          input              j                        ⟩                    .                      xe2x80x83                    ⁢                      P            u                                =                  "LeftBracketingBar"        Σ        "RightBracketingBar"            2      
In that case, the linearity of quantum mechanics gives an output state of the form       "LeftBracketingBar"    output    ⟩    =            ∑              j        =        1                    2        N              ⁢          xe2x80x83        ⁢                  a        j            ⁢                        ∑                      k            =            1                                2            N                          ⁢                              b            jk                    ⁢                                    "LeftBracketingBar"                              output                k                            ⟩                        .                              
The probability Pk of getting a specific output state k is then given by the square of its coefficient in the immediately preceding equation:       P    k    =      u    ⁢                  ∑                  j          =          1                          2          N                    ⁢              xe2x80x83            ⁢                        a          j                ⁢                  b          jk                ⁢                              u            2                    .                    
It can be seen that the probability of getting a particular output depends on all of the coefficients xcex2jk, which represent the results of all possible calculations on the computer. The result also depends on interference between all of the possible inputs, in the sense that Pk will be large if all of the input states contribute in phase with each other. Conversely, Pk will be small if the contributions from all of the initial states cancel out. The goal of quantum computing is to program the computer in such a way that the desired result occurs with high probability while all incorrect results occur with negligible probability.
To illustrate the usefulness of superposition states of this kind, suppose that we want to calculate the quantity Q,       Q    =                  ∑                  j          =          1                          2          N                    ⁢              xe2x80x83            ⁢                        e          ij                ⁢                  f          ⁡                      (            j            )                                ,
where ƒ(j) is a highly nonlinear function of j. The quantity Q corresponds to a weighted average of the function ƒ over all possible inputs to the computer, which is a Fourier transform of sorts. Calculations of this kind could be implemented on a quantum computer by programming the computer itself to calculate ƒ(j) and then creating a superposition of input states corresponding to the desired weighted average.
It has been shown that quantum computers could be used to efficiently factor large numbers, which is responsible for much of the current interest in quantum computing. The algorithm involved uses interference effects to ensure that, with high probability, the output of the computer will correspond to one of the desired factors.
Any practical implementation of a quantum computer will probably require a modular approach in which many separate logic gates can be connected with some equivalent of the wiring in a conventional computer. The ability to correct for the growth of errors in the quantum states, known as decoherence, is also essential. Individual quantum gates have been demonstrated using the nuclear spins of ions in a trap. This approach is not modular, however, and the transfer of information from one ion to another is a very complex process.
An optical approach to quantum computing appears to offer a number of practical advantages. All quantum computers are inherently dependent on interference effects and must maintain the appropriate phases. Optical interferometers are widely used in many current applications because their phase is relatively stable and can be controlled using feedback techniques. Interferometers based on charged particles, such as electrons, do exist but are very sensitive to stray electromagnetic fields. In addition, optical fibers or waveguides could readily be used to connect optical quantum gates as needed to perform the desired logic operations. For these and other reasons, the most practical approach to the construction of quantum computers will likely be based on the use of optical devices.
The primary difficulty in such an optical approach is that nonlinear effects of this kind typically require high-intensity electric fields, whereas the electric field associated with a single photon is normally quite weak. However, the field from a single photon is inversely proportional to the square root of the volume that it occupies, and confining a photon to a sufficiently small volume can produce electric fields as high as 10,000 V/m. Nonlinear phase shifts of this kind at the two-photon level have been demonstrated, but the approach involves the use of extremely high-quality mirrors, atomic beams, and operation near the resonant frequency of the atoms in the medium, none of which appear to be practical for the construction of a working quantum computer.
It was recently shown that any logic operation or numerical calculation can be implemented by combining a sufficient number of the controlled-NOT (XOR) gates illustrated in FIG. 3 with additional single-bit operations that are easily implemented. The controlled NOT has two binary inputs, A and B. Input A is always transferred to the output without change, while input B is inverted (flipped) if and only if input A=1. Thus, input A can control what happens to input B. The development of a practical controlled-NOT gate is the first step toward the construction of a quantum computer.
A controlled-NOT gate can be implemented using the optical arrangement illustrated in FIG. 4. Here, bit A has the value 1 if a single photon is in the path indicated by the dashed line, whereas it has the value 0 if that photon is in the path indicated by the solid line. Input B is represented in a similar way by a second photon; the two photons have different frequencies xcfx891 and xcfx892, which allow them to be distinguished. The two paths for photon B are combined by a beam splitter to form an interferometer with one arm passing through a nonlinear medium. The phase shift experienced by photon B depends on the index of refraction of the medium, which in turn depends on the strength of the electric field at that location (Kerr effect). If photon A passes through the medium at the same time, its electric field will introduce an additional xcfx80 phase shift, which changes the output path that photon B must take. The net result is that photon A can control the path of photon B.
The approach of the invention is based on a new physical effect that should greatly enhance these kinds of nonlinear phase shifts. Earlier nonlinear mechanisms involved the interaction of two photons with individual atoms, which gives a phase shift proportional to the number NA of atoms in the medium. The new mechanism involves the interaction of two photons with pairs of atoms, which gives a phase shift proportional to NA2, since that is the number of pairs of atoms in the medium. As FIG. 5 shows, the proposed mechanism consists of the absorption of photon 1 and the emission of photon 2 by atom A, followed by the absorption of photon 2 and the emission of photon 1 by atom B. (The energy of a quantum-mechanical system is uncertain over small time intervals and need not be conserved during the intermediate steps of this process.) This exchange of the photons by a pair of atoms has no net effect other than to cause a shift in the energy of the system, which produces the desired phase shift.
For large values of NA, this new mechanism should produce much larger phase shifts at the two-photon level. This in turn will allow other design requirements to be relaxed, such as the need for high-quality mirrors or atomic beams. As a result, this approach is eventually expected to allow the construction of large numbers of quantum gates on a single substrate, with optical waveguides to provide the necessary logical connections.
The main advantages of the invention over other techniques are:
independent logic gates;
ability to make connections between the independent logic gates using optical fibers or waveguides;
low error rate (decoherence) due to the ability to operate at large frequency detunings;
potential ability to fabricate large numbers of logic devices on a single substrate using optical waveguides and micro-fabrication techniques, in analogy with semi-conductor technology;
high rate of logic operations due to the propagation of information at the speed of light; and
compensation for the effects of dispersion.
As a result of these advantages, the method of the invention is expected to provide a practical means of scaling-up to a full-size computer. Furthermore, the method disclosed herein can be applied to conventional optical data processing, i.e., the optical approach described above can be used to build a standard computer to increase speed and reduce heat generated by the components.